Problem Situation:A construction company wishes to build a rectangular enclosure to store machinery and equipment. The site selected borders on a river that will be used as one of the sides of the rectangle. Fencing will be needed to form the other three sides. The company has 682 feet of 10-foot high chain-link fencing. The questions that follow should help you determine the dimensions of the rectangle that will produce the maximum area for the storage site. Assume that the company wishes to use all of the fencing, and that “width” refers to the measure of each of the two sides that are perpendicular to the river, and “length” refers to the measure of the side parallel to the river (see diagram). For the questions that follow, all measurement answers are to be given with the correct unit label, e.g. feet, square feet, etc.

All of the questions are in the file

4/08

MATH 145

Project for Unit 3

Maximum Storage Area

DUE DATE:

This project is worth 10% of your Unit 2 grade. Please review the Project FAQ handout for

format and process.

Problem Situation:

A construction company wishes to build a rectangular enclosure to store machinery and

equipment. The site selected borders on a river that will be used as one of the sides of the

rectangle. Fencing will be needed to form the other three sides. The company has 682 feet of 10foot high chain-link fencing. The questions that follow should help you determine the

dimensions of the rectangle that will produce the maximum area for the storage site. Assume

that the company wishes to use all of the fencing, and that “width” refers to the measure of each

of the two sides that are perpendicular to the river, and “length” refers to the measure of the side

parallel to the river (see diagram). For the questions that follow, all measurement answers are to

be given with the correct unit label, e.g. feet, square feet, etc.

River

width

Storage Area

width

length

PART 1

1. Copy the following table into your project and complete it, showing the dimensions (width

and length) of some possible enclosures along with the resulting areas. (Recall that the company

is using 682 feet of fencing total.)

Width (feet)

Length (feet)

Area (sq. ft.)

25

632

15,800

50

582

29,100

75

100

200

225

250

275

2. Of all the dimensions (length and width) listed in the table, which choice gives the largest area

for the enclosure? Express your answer with appropriate units.

3. Let w represent the width, and l represent the length. Write an equation expressing l in terms

of w.

4. Write the general formula for the area A of a rectangle in terms of l and w.

5. Use the general formula to write the area A of the enclosure as a function of w (that is, only the

variable w should appear in the function). Write the function in simplest form. What type of

function is this?

6. Use your calculator to obtain a graph of the area function. Choose a window that shows both

horizontal intercepts and the vertex. Locate the point on your graph that represents your answer

to Question 2 by letting Y2 = the corresponding area, then finding the intersection. (Note there

will be two intersections, you must choose the correct one). Draw an arrow pointing to the

point that represents your answer to Question 2, and write the coordinates as an ordered

pair.

Go to the Math Lab or library and either obtain a printout of the screen or copy it to the clipboard

and paste it into your WORD document. In either case, label the axes with variable and word

labels, and label the tic marks on each axis. Look at the window on your graphing calculator to

help with this. Label the graph with its function. Give your graph a title. Write the window

settings you used: Xmin, Xmax, Xscl, Ymin, Ymax, Yscl.

7. What do we call the graph of this area function?

8. What is the name of the point that represents the maximum area?

9. Find the width needed to give the maximum area algebraically. Show the formula you used

and the steps to the solution. Express your answer with appropriate unit.

10. Now that you know the width needed to find the maximum area, find the corresponding

length of the enclosure of maximum area using the formula from question 3. Show the steps to

your solution. Express your answer with appropriate unit.

11. Now find the maximum area algebraically using the function A( w) . Express your answer

with appropriate unit.

PART 2

12. The width, w, is the input for this area function. Does w = 350 have any practical meaning in

this situation? Explain showing an appropriate calculation.

13. Use an algebraic method to find the horizontal intercepts of this area function. Show the

equation you solved and how you solved it.

PART 3

Instead of building the enclosure of maximum area, the company wishes to build only a 50,000

square foot enclosure still using up all the fencing and situated with one side on the river. Find

the dimensions of the enclosure that will satisfy these requirements by doing the following:

14. What equation can be solved to find the possible widths?

15. Use your calculator to solve this equation graphically. Obtain a printout, label each graph

with its equation, write down the window used, draw arrows pointing to the solutions.

Write your solutions to the nearest tenth of a foot and express using appropriate units

16. Using the widths from Question 15, find the corresponding lengths. Express your answers to

the nearest tenth of a foot, and with appropriate units.

17. In sentence form, tell the company the dimensions (length x width) that it can use to satisfy

its requirements.

18. Now support your answers that you obtained graphically by showing how to solve the

equation in problem #14 algebraically. Show all steps in the solution and write the resulting

widths to the nearest tenth of a foot and with appropriate unit.